Question

What is the value of ${{\left[ {{\log }_{10}}\left( 5{{\log }_{10}}100 \right) \right]}^{2}}$?
(a) $4$
(b) $3$
(c) $2$
(d) $1$

Answer 1

Hint: In the above question, we have been given a logarithmic expression whose value has to be obtained, which is given as ${{\left[ {{\log }_{10}}\left( 5{{\log }_{10}}100 \right) \right]}^{2}}$. For this, we need to start simplifying the innermost logarithmic term, which is ${{\log }_{10}}100$. This can be simplified by putting $100={{10}^{2}}$ and using the logarithmic property given by ${{\log }_{a}}{{a}^{m}}=m$. Then we will be left with a single logarithmic term which can be simplified similarly. Finally taking the square of the simplified expression obtained, we will be able to obtain the correct answer.

Complete step by step solution:
The expression given in the above question is
$\Rightarrow E={{\left[ {{\log }_{10}}\left( 5{{\log }_{10}}100 \right) \right]}^{2}}$
We start by simplifying the innermost logarithmic term, ${{\log }_{10}}100$. For this, we put $100={{10}^{2}}$ in the above expression to get
$\Rightarrow E={{\left[ {{\log }_{10}}\left( 5{{\log }_{10}}{{10}^{2}} \right) \right]}^{2}}$
Now, by using the property of the logarithmic function given by ${{\log }_{a}}{{a}^{m}}=m$, we can simplify the above expression as
$\begin{align}
  & \Rightarrow E={{\left[ {{\log }_{10}}\left( 5\times 2 \right) \right]}^{2}} \\
 & \Rightarrow E={{\left[ {{\log }_{10}}10 \right]}^{2}} \\
\end{align}$
Now, we put $10={{10}^{1}}$ in the above expression to get
\[\Rightarrow E={{\left[ {{\log }_{10}}{{10}^{1}} \right]}^{2}}\]
Now, by again using the logarithmic property ${{\log }_{a}}{{a}^{m}}=m$, we can simplify the above expression as
\[\begin{align}
  & \Rightarrow E={{\left[ 1 \right]}^{2}} \\
 & \Rightarrow E=1 \\
\end{align}\]
Thus, the value of the given logarithmic expression is found to be equal to \[1\].
Hence, the correct answer is option (d).

Note: For solving these types of questions, we have to remember all the important properties of the logarithmic function. We may apply the logarithmic property given by $\log AB=\log A+\log B$ to simplify the given expression as ${{\left[ {{\log }_{10}}5+{{\log }_{10}}100 \right]}^{2}}$. But that will only increase our calculations to get the final answer.